This is a good question. I'd like to rephrase your question as, "How did the U.S. government finance the war effort of 1861-65"?

The answer is, not much differently than the government financed the Second World War. Here are three articles on the topic --

http://www.enotes.com/major-acts-congress/national-bank-act
http://minerva.union.edu/kleind/eco024/documents/bank/national_currency_act.htm
http://acct.tamu.edu/giroux/financingcivil.htm

As best I can understand, the IRS paid the interest on war bonds issued by the government. The IRS was abolished in 1872.